Trying to compute the inequality in Cartesian coordinates is an efficient way to make yourself crazy.

It may be more helpful to use a geometrical interpretation of the equation first in the Argand diagram. Keep in mind that | z - z0 | is the "length" of a vector from the point representing z0 to the point representing z . The equation | z - 2i | = | z + i | then describes the curve in the Argand diagram of points equidistant from 2i and -i . What is that curve like? The inequality then represents the set of points closer to 2i than to -i . Where is that region?